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MATH 370 Final Review

MATH 370 Final Review. Chapter 5-10. Know this well. Chapter 5 and 6: Counting/ Probability Basic counting P( n,r ), C( n,r ), C(n-1+r,r),… Basic probability, expected value Ch . 7: Recurrence Finding solutions Proving these are solutions Ch . 8: Relations Relation, function defs

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MATH 370 Final Review

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  1. MATH 370 Final Review Chapter 5-10

  2. Know this well Chapter 5 and 6: Counting/ Probability • Basic counting • P(n,r), C(n,r), C(n-1+r,r),… • Basic probability, expected value Ch. 7: Recurrence • Finding solutions • Proving these are solutions Ch. 8: Relations • Relation, function defs • Def of R,S, A, T • Def of divides • Def of a=b mod m… • Def of Equiv Relation: RST • Def of PO: RAT • Def of comparable • Def of total order

  3. …topics to know well Ch. 9: Graphs • Special graphs: Kn, Cn, Wn, Qn, Km,n • Prove: • Bipartitite or not • Isomorphic or not • Planar or not • 9.8: Thm 1- chromatic # of planar graph ≤4 Ch. 10: Trees • Def of tree, rooted tree Basic Proof Methods • Direct proofs, utilizing definitions (ex: show R is transitive using the definition– Assume aRb and bRc. Show aRc.) • Indirect (contrapositive) and By Contradiction • Cases • Induction • Disproving, by using a counterexample

  4. Techniques to apply You won’t need to state these definitions, but be able to do: Ch. 7: • 7.5: |AUAUA|=… Ch. 8: • Matrices and digraphs and graphs • Do relations have certain properties: RSAT • Find closures • For (a,b) R4, find path length 4 in R • Maximal, minimal, greatest, least, glb, lub • Hasse • Compatible total order

  5. …Techniques to apply Ch. 9: • Thm. 2: undirected graph has an even # of odd degree • Calculate deg, deg-, deg+ • Adjacency tables and matrices • Paths • Strong and weakly connected • Counting paths of length l • Euler and Hamilton paths and circuits • Conditions for Euler paths and circuits (not for Hamilton) • Chromatic number of special graphs Ch. 10: • Determine if a tree or not • 10.3: pre, in, and postorder and Infix, prefix, postfix notation • 10.4: find spanning tree • 10.5: find minimum spanning tree

  6. Will be provided, so you can use them to calculate/ prove other things Ch. 5 • Binomial formula Ch. 7 • 7.1: ∑ari= … • 7.2: Thm 1 and 2 on how to find solutions to recurrence relations • N(P1’P2’…) formula • SοR definition; R n+1 = R n ο R; M S ο R = M R ο M S Ch. 8 • These statements will be given, so you may need to prove them: • 8.1: Thm. 1. R is transitive implies R n R • Thm. 2: 8.4: R* = U R n is the transitive closure of R • R* is transitive

  7. …Will be provided, so you can use them to calculate/ prove other things Ch. 9: • 9.2: Thm. 1 Handshaking: 2e= sum of deg(v)… • Euler: r=e-v+2 • Cor 1: If G connected, planar, simple, e≤ 3v-6 • Cor3: If G conn, planar simple, with no circuits length 3, then e≤2v-4 • Thm. 2: A graph is nonplanariff it contains a subgraphhomeomorphic to K3,3 or K5. Ch. 10: • 10.1: Thm 2: tree with n vertices has n-1 edges • Thm 3: full m-ary tree with I internal vertices contains n=mi+1 vertices • 10.1: Thm 4 (p.691): A full m-ary tree with n vertices has i=(n-1)/m internal vertices…

  8. Skip these • Ch. 6: Derangements • 8.2: databases • Any proofs in 9.6 • Proofs in ch. 10

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